## Paper on critical temperature revised

06/07/2011

Yesterday, I have uploaded a new version of my paper on the critical temperature of chiral symmetry breaking in QCD (see here). The reason for this was that there are some points in need for a better clarification. The main of these is the mapping theorem: I have added a sketch of a proof. The reason for this is that there is a common misunderstanding about it and that some people think  that this theorem is for quantum field theories. Indeed, it just establishes a map between classical solutions of a scalar field and a Yang-Mills field but in the asymptotic limit of a coupling going to infinity. Quantum theory does not enter at all here but these classical asymptotic solutions can be used to build up a perturbation theory for quantum field theory in the infrared, that is for low-energies, that is the range of interest for all the phenomenology we would like to understand.

Another recurring question is if this mapping breaks in some way gauge invariance. The answer is a resounding no as the proof does not select a gauge at the start but anyhow if one wants quantization a gauge must be selected.

Finally, I have better clarified the derivation of the critical temperature and added some more relevant references. I hope in this way that my arguments can be better understood. Indeed, presentation is one of the most difficult aspects of scientific communication and sometime it is a sound explanation of attrition between authors and referees.

## A critical point in QCD exists indeed!

30/05/2011

After my comeback from the conference in Ghent (see here, here and here), I started a collaboration with Marco Ruggieri. Marco was instrumental in making me aware of that part of the community that does computations in QCD at finite temperature. The aim of these people is to get a full landscape of the ground state of hadronic matter, even when a magnetic field is applied and the vacuum state is expected to change. This is not just an intellectual exercise as recent observations of quark-gluon plasma are there to show and also some important experiments at LHC are now unfolding the complexity of this theory. So, we are saying about the forefront of modern research in the field of nuclear matter that can have significant impact in our understanding of the early universe.

As Marco pointed out in this blog (see here), due to the lack of knowledge of techniques to manage QCD at low-energies, we are not even able to give a definite answer to the question if different phases exist for hadronic matter and if a critical temperature, or a cross-over temperature, can be found from a theoretical standpoint. People use two different approaches to manage this question: lattice computations and phenomenological models like Nambu-Jona-Lasinio or sigma models. Lattice computations displayed a critical temperature at zero quark masses and zero chemical potential  (see here) and a cross-over rather than a phase transition with non-zero quark masses. A critical temperature was found to be about 170 MeV. These studies are yet underway and improve year after year. From a theoretical point of view the situation is less clear even if a Nambu-Jona-Lasinio model can be used to work out a critical temperature. The model should be non-local.

With this scenario in view, it seems not thinkable a proof of existence of a critical point at zero quark masses and zero chemical potential. This is true unless we know how to manage the low-energy behavior of QCD. One should have solved the mass gap problem to say in a few words what is needed here. As my readers know, I am in a position to give a definite answer to such a question and, of course I did. On Friday I have uploaded a new paper of mine on arXiv (see here) and I have obtained an evidence for a critical point in QCD. This is the point in temperature where the chiral symmetry gets restored.

The idea for this paper come after I read a beautiful work by T. Hell, S. Roessner, M. Cristoforetti, W. Weise on the non-local Nambu-Jona-Lasinio model (see here). These authors completely work out the physics of this model. The point is that, as I have shown, this is the right one to describe low-energy physics in QCD. From a comparison with the form factors, mine obtained solving QCD with the mapping theorem and the one of Weise&al. guessed from a model of a liquid of instantons, the agreement is so good that my approach strongly supports the other view.

Finally, I was able to get the long sought equation for the critical temperature at zero quark masses and chemical potential. At this temperature the chiral symmetry appears to be restored. I find really interesting the fact that a similar equation was obtained by Norberto Scoccola and Daniel Gomez Dumm (see here). My equation for the critical temperature is substantially the same  as theirs. Of course, the fundamental difference between my approach and all others relies on the fact that I am able to get the form factor solving QCD. In the preceding works this is just a guess, even if a very good one. Besides, so far, nobody was able to show that a Nambu-Jona-Lasinio model is the right low-energy limit of QCD. This result should be ascribed to me and Ken-Ichi Kondo (see here).

Having proved that a non-local Nambu-Jona-Lasinio model is the right low-energy limit, a prove of existence of a critical point is so obtained. This proof will be presented at the next conference in Paris  on non-perturbative QCD (see here). Me and Marco will be there the next week.

Z. Fodor, & S. D. Katz (2004). Critical point of QCD at finite T and \mu, lattice results for physical
quark masses JHEP 0404 (2004) 050 arXiv: hep-lat/0402006v1

Marco Frasca (2011). Chiral symmetry in the low-energy limit of QCD at finite temperature arXiv arXiv: 1105.5274v2

T. Hell, S. Roessner, M. Cristoforetti, & W. Weise (2008). Dynamics and thermodynamics of a nonlocal Polyakov–Nambu–Jona-Lasinio
model with running coupling Phys.Rev.D79:014022,2009 arXiv: 0810.1099v2

D. Gomez Dumm, & N. N. Scoccola (2004). Characteristics of the chiral phase transition in nonlocal quark models Phys.Rev. C72 (2005) 014909 arXiv: hep-ph/0410262v2